CHAPITRE 4 Value at Risk

CHAPITRE 4
Value at Risk
Michel LUBRANO
Avril 2011
Contents
1
Introduction
1.1 VaR d’une position longue . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
2
Risk and volatility
4
3
Estimation methods for VaR
3.1 Historical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Modeling the dependence of returns . . . . . . . . . . . . . . . . . . . . .
4
4
4
5
4
Engle and Manganelli: CAViaR 2004
5
5
Backtesting VaR estimates
5.1 Likelihood ratio test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 CAViaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Hurlin 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
6
7
7
6
References
7
1
1 INTRODUCTION
2
1 Introduction
1. La VaR d’un portefeuille d’actifs financiers est une mesure quantitative de risque
attachée à la valeur du portefeuille.
2. Si un investisseur détient un portefeuille (une position longue), il encourt le risque
d’une perte de valeur sur un certain horizon.
3. Les accords de Bâle sur la réglementation des banques requièrent que celles-ci constituent des provisions de capital pour couvrir ce type de risques. Les banques doivent
elles-mêmes évaluer la VaR de leurs actifs financiers. The Basel Committee on Banking Supervision (1996) at the Bank for International Settlements uses VaR to require
financial institutions such as banks and investment firms to meet capital requirements
to cover the market risks that they incur as a result of their normal operations.
4. Une banque pourrait dire que la VaR de son portefeuille, à un certain horizon, est de
20 millions d’euros au niveau de confiance de 95% (ou au niveau de risque statistique
de 5%). Autrement dit, il y a 5 chances sur 100 qu’une perte supérieure à 20 millions
se produise à l’horizon considéré, si les marchés fonctionnent normalement.
5. Comment peut-on estimer ce montant? Est-il de 20, 5, ou 50 millions? La VaR
est définie pour un niveau de risque (comme 5%), un horizon de temps (comme 10
jours), et sur la base d’un modèle d’évaluation du risque financier.
6. La VaR est déduite directement d’un quantile de la distribution (conditionnelle) du
rendement du portefeuille.
1.1 VaR d’une position longue
Soit yt le rendement aléatoire, à une période, d’un portefeuille qui vaut pt :
yt = 100∆ log pt .
Soit qt (α) le quantile de niveau α% de la distribution de yt , défini par la relation
Pr[yt < qt (α)] = α.
La VaR de niveau α de pt est la perte minimale qui peut se produire avec une probabilité
égale α:
V aRt (α) = −qt (α) × pt .
Par exemple, si
yt ∼ N(µt , σt2 ),
V aRt (α) = −(µt + zα σt )pt ,
où zα est le quantile de niveau α de la distribution N(0,1).
1 INTRODUCTION
3
Exemple de VaR (distribution normale)
N(0.0354,1.1972)
0.30
0.25
0.20
0.15
0.10
0.05
q(0.05) = −1.94
α= 0.05
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
V aR(0.05) = −(0.0354
| {z } −1.645
| {z })pt = 1.94 pt ,
| {z } × 1.197
µ
σ
z0.05
Chaire Francqui 0 – p. 43/97
Exemple de VaR (distribution empirique)
0.5
Empirical density of Nasdaq daily returns
0.4
0.3
0.2
0.1
q(0.05) = −1.89
α= 0.05
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
On a observé un rendement inférieur à -1.89% dans 5% du
nombre total de jours de l’échantillon =⇒ V aR(0.05) = 1.89 pt .
Chaire Francqui 0 – p. 44/97
Figure 1: VaR du Nasdaq
2 RISK AND VOLATILITY
4
2 Risk and volatility
On peut mesurer la volatilité comme
Vθ = Et−1 [|Rt − µt |θ ]
Mean absolute deviation or variance. Variance adequate only under quadratic utility or normal distribution of returns.
Risk only exists in the left part of the distribution. We do not want to diversify to reduce
the possibility of an unexpected positive return.
VaR
V aRtα = −sup[r|Prt−1 [Rt ≤ r] ≤ α]
Basel
1. α = 0.01
2. 10 trading days
3. one year for backtesting
3 Estimation methods for VaR
3.1 Historical simulation
The VaR is estimated it as the α th quantile of the empirical distribution of losses:
V̂ aRtα = −Rω:T ,
where Rω:T is the αth-order statistic of the data and
ω = [T α] = max{m|m ≤ T α, m ∈ IN}.
But the empirical quantile is not a good estimator for an extreme quantile.
As seen in the seminar, cannot be revised daily. Because we need a lot of observations
to estimate the quantiles.
3.2 Bootstrap
If IID returns, bootstrapping in the empirical distribution of returns. HS estimation using
the replications.
4 ENGLE AND MANGANELLI: CAVIAR 2004
5
3.3 Modeling the dependence of returns
Returns modeled as
Rt = µt + ǫt σt
Then
V aRtα = µt + qα σt
The mean is modeled using an ARMA
µt = φ0 + φ1 Rt−1 + θ1 at−1
at = Rt − µt
The variance is modeled using a GARCH
2
σt2 = α0 + α1 a2t−1 + βσt−1
Finally, the quantile can be determined easily if the distribution of ǫ is normal. Other
distributions can be used when estimating the GARCH like the Student of asymmetric
Students.
4 Engle and Manganelli: CAViaR 2004
VaR is an estimate of how much a certain portfolio can lose within a given time period,
for a given confidence level. The great popularity that this instrument has achieved among
financial practitioners is essentially due to its conceptual simplicity: VaR reduces the (market) risk associated with any portfolio to just one monetary amount.
Despite its conceptual simplicity, the measurement of VaR is a very challenging statistical problem and none of the methodologies developed so far gives satisfactory solutions.
Since VaR is simply a particular quantile of future portfolio values, conditional on current
information, and since the distribution of portfolio returns typically changes over time, the
challenge is to find a suitable model for time varying conditional quantiles.
1. provide a formula for calculating V aRt as a function of variables known at time t − 1
and a set of parameters that need to be estimated
2. provide a procedure (namely, a loss function and a suitable optimisation algorithm)
to estimate the set of unknown parameters
3. provide a test to establish the quality of the estimate.
Some first estimate the volatility of the portfolio, perhaps by GARCH or exponential
smoothing, and then compute VaR from this, often assuming normality.
The volatility approach assumes that the negative extremes follow the same process as
the rest of the returns and that the distribution of the returns divided by standard deviations
5 BACKTESTING VAR ESTIMATES
6
will be independent and identically distributed, if not normal.
CAViarR: model directly the quantile of the distribution of returns. We want the quantile to evolve smoothly: so autoregressive. We want explanations: the x.
α
α
V aRtα = β0 + β1 V aRt−1
+ l(β2 , Rt−1 , V aRt−1
)
where different forms for the function l were proposed.
1. symmetric slopes in R
+
−
2. asymmetric slopes l = β2 Rt−1
+ β3 Rt−1
α
3. adaptative l = β0 ([1 + exp(G[Rt−1 + V aRt−1
])]−1 − α)
G>0
Estimation using the methods of the quantile regression.
5 Backtesting VaR estimates
Imposed by Basel Committee on Banking Supervision. Backtesting over one year (250
obs). Backtesting is based on a failure process
Itα = 1I(Rt < −V aRtα ),
t = T + 1, ..., T + n
One year ahead for testing, n = 250. A VaR estimate is accurate iff
Et−1 [Itα ] = α.
5.1 Likelihood ratio test
The number of failures x has a binomial distribution with parameters α and n. Kuipec
(1995)
"
LRuc = 2 log
x
1−
n
n−x x #
x
n
− 2 log[(1 − α)n−x αx ] ∼ χ2 (1).
χ2 (1) = 3.84 at 95%. Low power in small samples.
We could devise a test for
α
Et−1 [Itα |It−1
] = α.
It is a test of independence. LRind . The test for conditional coverage is
LRcc = LRuc + LRind ∼ χ2 (2)
χ2 (2) = 5.99 at 95%.
6 REFERENCES
7
5.2 CAViaR
A dynamic quantile test based on a regression of Itα − α on its lags and the other values
contained in the conditioning set. See if these variables are significant.
5.3 Hurlin 2011
Backtesting Value-at-Risk : A GMM Duration-based Test. The duration between hits is
a geometric distribution. Using properties of this distribution, Hurlin finds polynomials
which are zero under the null. The first polynomial corresponds to the first moment. Test if
these polynomials are jointly zero. Has to determine the number of polynomials: 1, 2 or 3.
6 References
Engle, R. and S. Manganelli (2004) Caviar: Conditional autoregressive value at risk by
regression quantiles. Journal of Business and Economic Statistics 22, 367-381.
Kupiec, P. (1995). Techniques for verifying the accuracy of risk management models.
Journal of Derivatives 3, 73-84.
Marı́a Rosa Nieto and Esther Ruiz (2008) Measuring financial risk: comparison of alternative procedures to estimate VaR and ES. DP Carlos III, Madrid.
Basel Committee on Banking Supervision (1996a). Supervisory framework for the use
of backtesting in conjunction with the internal models approach to market risk capital
requirements, bank for international settlements. Technical report.